292
An Introduction to Beam Physics
Furthermore, we obtain that
S
−
32 + S
+
31 = −2T S sin μ, S
−
32 − S
+
31 = −2iT S cos μ.
Hence we arrive at the conclusion that T S is real. Therefore, we have
N 3 =
e
−iμ
1 − iT S s
−
0 s
+
0
s
−
0
e
iμ
1 + iT S s
−
0 s
+
0
s
+
0
.
To the first order of T S , N 3 can be expressed as
N 3 =
exp
−i
μ + T S s
−
0 s
+
0
s
−
0
exp
i
μ + T S s
−
0 s
+
0
s
+
0
.
It is clear by far that the remaining terms in the normalized map N 3 contribute
to the change of the tune only. It is worth noting that the change of the tune
is a function of the invariant, which is sometimes called the tune shift with
amplitude. Computationally, the above described procedure can be easily
carried out using the Differential Algebraic (DA) technique, which is valid for
any given order.
Now the tune shift with amplitude can be determined through the relations
between the coefficients in the real and the complex coordinates. Repeating
eq. (11.13) to the third order, using eqs. (11.11) and (11.12), we have
s
−
1
s
+
1
=
1
2
1 i
1 −i
cos μ sin μ
− sin μ cos μ
1 1
−i i
s
−
0
s
+
0
+
1
4
1 i
1 −i
A 111
s
−
0 + s
+
0
3 + A 112
s
−
0 + s
+
0
2 −is
−
0 + is
+
0
B 111
s
−
0 + s
+
0
3 + B 112
s
−
0 + s
+
0
2 −is
−
0 + is
+
0
+
1
4
1 i
1 −i
A 122
s
−
0 + s
+
0
−is
−
0 + is
+
0
2 + A 222
−is
−
0 + is
+
0
3
B 122
s
−
0 + s
+
0
−is
−
0 + is
+
0
2 + B 222
−is
−
0 + is
+
0
3
=
e
−iμ 0
0 e
iμ
s
−
0
s
+
0
+
1
4
(A 111 +iB 111 )
s
−
0 +s
+
0
3 +(A 112 +iB 112 )
s
−
0 +s
+
0
2 −is
−
0 + is
+
0
(A 111 −iB 111 )
s
−
0 +s
+
0
3 +(A 112 −iB 112 )
s
−
0 +s
+
0
2 −is
−
0 + is
+
0
+
1
4
(A 122 +iB 122 )
s
−
0 +s
+
0
−is
−
0 +is
+
0
2 +(A 222 +iB 222 )
−is
−
0 +is
+
0
3
(A 122 −iB 122 )
s
−
0 +s
+
0
−is
−
0 +is
+
0
2 +(A 222 −iB 222 )
−is
−
0 +is
+
0
3
.
It is straightforward to extract the coefficients S
−
32 and S
+
31 , which are
S
−
32 =
1
4
[3 (A 111 +iB 111 )−i (A 112 +iB 112 )+(A 122 +iB 122 )−3i (A 222 +iB 222 )],
S
+
31 =
1
4
[3 (A 111 −iB 111 )+i (A 112 −iB 112 )+(A 122 −iB 122 )+3i (A 222 −iB 222 )].